Pappus' Generalization of the Pythagorean theorem
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This animation gives a visual proof of Pappus' generalization of
the Pythagorean theorem. (See notes below.)
Theorem (Pappus): Let
ABC be a triangle, and let parallelograms ABDE and ACFG be erected externally
to ABC with
respective bases AB and AC. Let H be the point where the sides DE and FG,
extended, of
these parallelograms meet. If BCJK is the parallelogram erected upon BC, external
to triangle ABC and having the sides CJ and BK parallel to
and congruent with the segment HA, then the area of BCJK is the sum of the
areas of ABDE and ACFG.
In this visual proof, the two original parallelograms deform so that they retain
their original bases and altitudes (because one side of each slides along one
member of a pair of parallel lines) until they meet at the lower left. These
deformations do not change the areas of the parallelograms.
Then they
deform together, again remaining between pairs of parallel lines, and again in
ways that do not alter their areas, until they fill a parallelogram on the
third side of the triangle. (12/01/07)